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For any cohomology theory $H$ that can be factorized through (theMorel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ weestablish the $SH^{S^1}(k)$-functoriality of coniveau spectral sequences for$H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousincomplex for $H^*(S)$ splits; if $H$ also factorizes through $SH^+(k)$ or$DM(k)$, then this is also true for any primitive $S$. Moreover, the cohomologyof such an $S$ is a direct summand of the cohomology of any its open densesubscheme. This is a vast generalization of the results of a previous paper.  In order to prove these results we consider certain triangulated categoriesof motivic pro-spectra, and introduce Gersten weight structures for them. Westudy in detail the notions of cohomological dimensions of scheme associated tovarious categories of motivic pro-spectra; they are defined in terms of thecorresponding weight structures and count the number of non-zero Nisnevichcohomology for sheaves in the hearts of orthogonal "homotopy" $t$-structures.

Gersten weight structures for motivic homotopy categories; retracts of  cohomology of function fields, motivic dimensions, and coniveau spectral  sequences